Did you mean det(A)= alpha
Hi, can someone help me out with this, how do I solve this matrix question:
Let A = [a,b,c; d,e,f; g,h,i] Given that detA= alpha and alpha is not equal to zero, express each of the following terms of alpha:
a) det(2A)
b) det(A^-1)
c) det [2a,2b,2c; d,e,f; g,h,i]
d) det [a,b,c; d,e,f; g,h,i]
e) det [a,b,c; d-3a, e-3b, f-3c; g,h,i]
Thank you
det(A) = a(ei - fh) - b(di-fg) + c(dh-eg) = alpha
Now:
det(2A) = 8a(ei - fh) - 8b(di-fg) + 8c(dh-eg) = 8(a(ei - fh) - b(di-fg) + c(dh-eg))= 8alpha
try the rest...
check Determinant - Wikipedia, the free encyclopedia
Use the property det(AB) = det(A).det(B)...
AB =
Let B =
we already know det(B) = alpha
Now find a matrix A multiplied by B,that will give you AB...
I will leave this to you to do, if you have trouble finding sucha matrix then post again.
Then Find det(A) multiply by det(B)...
I have not done this in a long time, so please excuse if there are any mistakes.
Hello, zzizi!
I'm a little stuck with this one. .How do I work out this determinant?
. .
How do you work out any determinant? . . . Just crank it out!
However, we can use some Properties on this one . . .
Here's an important property of determinants.
If a multiple of one row is added to (subtracted from) another row,
. . the value of the determinant is unchanged.
Suppose we have: . . . Its value is:. .[1]
Subtract row-2 minus row-1: .
Subtract row-3 minus twice row-1: .
This determinant has the same value . . . the value at [1].
Is the number one that you put in the bottom right hand corner of the determinant delibrate?
Suppose we have: . . Its value is:. (aei + bfg + cdh) - (afh + bdi + ceg) .[1]
And is the determinant alpha once again?
another way to see that subtracting 3 times row 1 from row 2 does not change the determinant:
let P be the matrix:
then
(verfiy this!)
since det(PA) = det(P)det(A), to find det(PA), we need to calculate det(P).
det(P) = (1)(1)(1) + (0)(0)(0) + (0)(-3)(0) - (0)(1)(0) - (0)(0)(1) - (1)(0)(-3) = 1 + 0 + 0 - 0 - 0 - 0 = 1
therefore, det(PA) = det(A) = α
(this was what Goku was hinting at his post).
you can see that the value of det(P) won't change if we replace "-3" by "r", and furthermore, it really doesn't matter "where" r goes (as long as it's not on the main diagonal), because there will always be some other 0 on any diagonal (or "extended diagonal", like when you "loop around") containing r to "cancel it out". so subtracting r times any row from a DIFFERENT row, will never change the determinant of a 3x3 matrix.